The Gunners had looked good value for a share of the spoils from the Group F encounter after recovering from a slow start - Henrikh Mkhitaryan put the well-drilled Germans ahead - to level before the break through Olivier Giroud.
Indeed, if a sublime 20-yard curling effort from substitute Santi Cazorla had flown just a couple of inches lower instead of crashing off the angle, Arsenal may well have recorded what would have been a hard-earned victory over last season's beaten finalists.
However, Arsene Wenger's side were left with the disappointing taste of defeat for only the second time this season when Robert Lewandowski - somewhat fortunate to still be on the pitch after earlier catching Laurent Koscielny with a stray elbow - converted following a clinical counter-attack in the 82nd minute.
Napoli's victory over whipping-boys Marseille leaves three teams now locked on six points and Arsenal now face a tough trip to Dortmund in a fortnight, where another loss would put their qualification hopes for the knockout stages in serious jeopardy.
Wilshere, though, insists Wenger's men have the strength of character to pick themselves up and go again when the Barclays Premier League leaders head to bottom club Crystal Palace for Saturday's lunchtime kick-off.
"It is never nice to lose at home, but it is not the end of the world," said Wilshere, who was substituted in the second half as a precaution after aggravating his ankle.
"We can't let this result affect us now, especially in the Premier League where we are on a great run,
"It will be tough away at Palace, it always is, they will fight for every ball and it will be a different game from Dortmund, but we still have to bounce back straight away."
Wilshere accepts Arsenal had been "nervous" during the opening exchanges against Jurgen Klopp's side, but sees no reason why they should not be confident of getting a positive result at Signal Iduna Park, where they drew 1-1 in September 2011 after being within two minutes of victory.
"We know what we have to do now," the England midfielder said.
"We aim to win every game, and it will be tough, but we have been to Dortmund before and were unlucky not to win on the night.
"We know we can go there to get something out of the game, and that is the aim.
"We will watch the game back as we always do and take a lot of positives out of it, we could have done better on their goal, but it was a great goal as they broke on us really quickly, with a good ball in and a great finish.
"But the good thing about this competition is you do not have to wait very long, in the next couple of weeks we will go out to Dortmund, and it will be all to play for."
Aaron Ramsey has been enjoying a fine season, but was robbed in possession on the edge of his own penalty area for the first goal and the Wales midfielder felt the whole team lost focus later in the match when they should have been concentrating on closing out a positive result.
"I felt like we were in control of the game, but then we switched off for 30 seconds, we were in their box and 30 seconds later the ball is in the back of our net," Ramsey said on Arsenal Player.
"We were in control of the game, we didn't look like we were going to concede, but then we did. That's what happens (when you switch off).
"I was disappointed as well with the way they scored their first goal, when I got caught on the ball.
"We showed good character getting back into game, and we created a few chances in the second half and hit the bar.
"On another night we may have taken the lead, but we have to learn from the way we defended for the second goal."